Local rotundity structure of Calderon-Lozanovskii spaces

General results saying that a point x of the unit sphere S(E) of a Kothe space E is an extreme point (a strongly extreme point) [an SU-point] of B(E) if and only if vertical bar x vertical bar is an extreme point (a strongly extreme point) [an SU-point] of B(E+) and vertical bar x vertical bar is a UM-point (a ULUM-point) [nothing more] of E are proved. These results are applied to get criteria for extreme points and SU-points of the unit ball in Calderon-Lozanovskii spaces which refer to problem XII from [5]. Strongly extreme points in these spaces are also discussed.